The number of molecules with energy greater than the threshold energy for a reaction increases five — Chemical Kinetics Chemistry Question
Question
The number of molecules with energy greater than the threshold energy for a reaction increases five fold by a rise of temperature from 27ºC to 42ºC. Its energy of activation in J/mol is ________. (Take ln 5 = 1.6 ; R = 8.3 J mol K ) –1 –1
💡 Solution & Explanation
**Step 1: Identify the relationship between molecular population and temperature** The fraction of molecules with energy ≥ Ea follows the Arrhenius distribution. When this fraction increases 5-fold, we use: $$\frac{N_2}{N_1} = 5 = e^{E_a/R(1/T_1 - 1/T_2)}$$ **Step 2: Convert temperatures to Kelvin** - T₁ = 27°C = 300 K - T₂ = 42°C = 315 K **Step 3: Take natural logarithm of both sides** $$\ln(5) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$ $$1.6 = \frac{E_a}{8.3}\left(\frac{1}{300} - \frac{1}{315}\right)$$ **Step 4: Calculate the temperature term** $$\frac{1}{300} - \frac{1}{315} = \frac{315 - 300}{300 \times 315} = \frac{15}{94500} = \frac{1}{6300}$$ **Step 5: Solve for Ea** $$1.6 = \frac{E_a}{8.3} \times \frac{1}{6300}$$ $$E_a = 1.6 \times 8.3 \times 6300$$ $$E_a = 13.28 \times 6300 = 83,664 \text{ J/mol}$$ Therefore, the answer is 83664.